AN APPLICATION OF THE CANONICAL CORRELATION

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1 AN APPLICATION OF THE CANONICAL CORRELATION Reader, Dan Petru Vasiliu, Ph. D Faculty of Economic tudies, University Titu Maiorescu Bucharest enior lect. Ganea Tudor, Ph. D Faculty of Economic tudies, University Titu Maiorescu Bucharest Abstract : the paper deals with an experimental application of the canonical correlation in the study of the social environment in the last year Romania. To do this, two groups of variables are selected : one group repersenting the social effort and the other the social effect of the labour. There studied the characteristic of the dependence between them. The conclusion is that in the considered period the greater was the effort, the smaller was the effect. Two different variants of the canonical correlation method were used to prove this Both of them confirm this conclusion. Keywords: canonical correlation ; canonical variates; the COREMAX method multivariate analysis The aim of the classical canonical correlation analysis is studying relationships between sets of multiple dependent and multiple independent variables. o, let X be a p dimensional and Y be a q dimensional random variable. We look for two vectors a, b such that a R coefficient correl ( a X,b Y be maximal. p,b R q and the quadratic KENDALL correlation Note that the univariate random variable U = a X and V = b Y are known as the canonical variates of the variables X, resp. Y,and that a, b are the corresponding canonical coefficients. This problem was solved long before (from Harold Hotteling -939 to Joseph F. Hair,998 for example. In this paper the canonical correlation method will be applied to study dependence between the group X = { civil employment ; employment rate } as the group of independent variables, and Y = { average net nominal monthly earnings ; total expenditure of household } as the group of independent variables. The data in our study are the next ( Year X X Y Y * * * *

2 Here, as stated above : X = civil employment X = employment rate Y = average net nominal monthly earnings Y = total expenditure of household. ( The stared data are author s estimations. First there proceeds to normalize this data,in order to obtain a less error level ( as is well known, the normalized variable V that corresponds to the variable V, is given by V V E(V σ (V (V =, where E(V is the expected value,and σ - the standard deviation of the variable V. The normalized variables are then represented by the next table : Year X X Y The estimated total correlation matrix of this last dataset is then : Y, with : 0,7834 0,7834 ; 0,967 0,6744 0,968 0,6783 0,967 0,968 0,6744 ; 0,6783 0,9957 0,9957 There determines now the largest eigenvalues of the matrices

3 A = ( and B = ( ( ( 0,9586 0,6 0,7034,5554 0,6959 0,0850 0,75,5770 The ( common largest eigenvalue of the matrices A, B is λ = 0,87. This is equally the quadratic measure of dependence between the two groups, { X,X } and { Y,Y } ( details will be seen later. The canonical coefficients are the eigenvectors, corresponding to λ = 0,87 : there obtain therefore a = ( 0,99 ; - 0,63, b = ( - 0,438 ; 0,904 U = 0,99 X - 0,63 X ; V = - 0,438 Y + 0,904 Y. U V If compute now the correlation coefficient correl (U,V, there obtains obviously, [correl (U, V ] = λ. correl (U,V = - 0,9385 ; For correl (U,V < 0, there results that the dependence of { Y,Y } with respect to { X,X } is of negative type : the greater { X,X } become, the poor { Y,Y } become. Otherwise, correl (U,V being so near of -, the dependence is strong enough to let us consider the group { X,X } determinative for { Y,Y }. The values of the canonical variates are processed below : X X Y Y

4 A reason the canonical correlation method is not used so widely is that the practical meaning of the canonical coefficients is rarely clear. To remediate that, different versions of the method are improved. One of them is the COREMAX method ( see (, that gives the coefficients the meaning of degrees of contributions of each component of the group to the corresponding canonical variate. To do this, we ll denote: η= α X µ = β Y + ( α X + ( β Y ; ; 0 α 0 β Now, we look for values α, β that maximize correl (η, µ. To make economic interpretation possible, we ll enforce to the variables another kind of being comparable, instead of normalization : thus, the normed variables are more adequate.this is justified by the fact that negative values of the variables are equally difficult to interpret ; or, the normalization unavoidable gives negative values to normalized variables. Remind that, for a variable series V, the normed series V is given by X X Y Y V V min{v} max{v} min{v} =. There obtaines :

5 Applying technique described in (, there obtains α = 0,47 ; β = 0,70 ; max Correl( µ ; η = - 0,8376 this value being close enough to max correl (U,V = - 0,9385. Finally : in the next, - the canonical variate µ = 0,47 X + 0,53 X gets the name of synthetical employment - the canonical variate η = 0,7 Y + 0,3 Y gets the name of synthetical earning Then: - the synthetical earning depends strongly from the synthetical employment ; - the intensity of this dependence is about 83,76% - the dependence is of the negative type : the greater the synthetical employment, the smaller the synthetical earning becomes. In other words : in the mentioned period, the greater was the effort, the smaller was the effect. BIBLIOGRAPHY: ( :Multivariate Data Analysis, 5th edition by Joseph F. Hair, Jr., Rolph E. u.a. Prentice Hall, Inc ( : Metode matematice pentru fundamentarea deciziilor in productie, by D.P.Vasiliu, Editura Tehnica, Bucuresti, 986

632 CHAP. 11 EIGENVALUES AND EIGENVECTORS. QR Method

632 CHAP. 11 EIGENVALUES AND EIGENVECTORS. QR Method 632 CHAP 11 EIGENVALUES AND EIGENVECTORS QR Method Suppose that A is a real symmetric matrix In the preceding section we saw how Householder s method is used to construct a similar tridiagonal matrix The